Σχηματισμός και εξισορρόπηση ζωνικών ανέμων σε πλανητικές τυρβώδεις ατμόσφαιρες

Transcription

1 Σχηματισμός και εξισορρόπηση ζωνικών ανέμων σε πλανητικές τυρβώδεις ατμόσφαιρες Ναβίτ Κωνσταντίνου και Πέτρος Ιωάννου Τμήμα Φυσικής Εθνικό και Καποδιστριακό Παν/μιο Αθηνών Πανεπιστήμιο Κύπρου 7 Ιανουαρίου 2014

2 N. Constantinou, U.o. A. Coherent structures in turbulent flows ocean currents NASA/Goddard Space Flight Center

3 Earth s atmospheric polar jet stream polar front jet NASA/Goddard Space Flight Center airplane trip from L.A. to Tokyo

4 striped Jupiter banded Jovian jets NASA/Cassini Jupiter Images observed Jovian zonal winds at cloud level Vasadava & Showman, 2005

5 N. Constantinou, U.o. A. Jet emergence on a barotropic beta-plane ψ(x, y, t), U (y, t) ψ(x, y, t) , U (y, t) y " = 2 3 " Kf /r x y " = = /(Kf r) = x = 6

6 Turbulent flows organize into jets Numerical simulation (barotropic beta-plane)

7 Jets seem to emerge as a bifurcation 0.7 zmf = E m/(e m + E p) " = " = ε E m : zonal energy, E p : eddy energy

8 Classical hydrodynamic stability hydrodynamic instabilities provide a way for eddies to gain energy from mean flow how about the opposite? Lord Rayleigh can the mean flow gain energy from the eddies through an instability?

9 Turbulence (usually) acts as a drag wall-bounded flow airflow over vehicle airflow over airfoil can turbulence act to reinforce large scale flows?

10 Our model: Barotropic vorticity equation on a t + u@ x + v@ y + v = r + p " f r ~u =0 ) u = v y dissipation stochastic x y u =

11 Zonal - Eddy field decomposition '(x, y, t) = (y, t)+' 0 (x, y, t) zonal mean eddy where (y, t) ='(y, t) = 1 L x Z Lx 0 '(x 0,y,t)dx 0

12 NL (nonlinear) t U = v 0 0 t 0 = U@ x 0 ( U yy )v 0 r 0 + F e + p " f F e = h y (v 0 0 y (v 0 0 x (u 0 0 ) eddy-eddy interaction term

13 QL (quasi-linear) t U = v 0 0 t 0 = U@ x 0 ( U yy )v 0 r 0 + F e + p " f F e = h y (v 0 0 y (v 0 0 x (u 0 0 ) eddy-eddy interaction term

14 QL (quasi-linear) t U = v 0 0 t 0 = A(U) 0 + p " f where A(U) = U@ x ( U yy )@ x 1 r

15 QL does NOT include turbulent cascades wave breaking nonlinear vorticity mixing The great wave off Kanagawa by Hokusai

16 QL captures the NL dynamics 6 (x, y, t) NL q(x, y, t), ϵ/ϵ c = (x, y, t) QL q(x, y, t), ϵ/ϵ c = y 3 y NL vorticity snapshot x mean flow comparison U (y) x QL vorticity snapshot

17 Our goal While QL captures and elucidates the jet-eddy dynamics it does not provide a predictive theory. Can we construct a theory that: Predicts when organized flows will emerge / describes jet formation as a bifurcation. Predicts the structure and the stability of the emergent zonal flows. Describes the jet merger dynamics???

18 Our goal While QL captures and elucidates the jet-eddy dynamics it does not provide a predictive theory. Can we construct a theory that: Predicts when organized flows will emerge / describes jet formation as a bifurcation. Predicts the structure and the stability of the emergent zonal flows. Describes the jet merger dynamics Such a theory can be constructed. It is based on the statistical dynamics associated with the QL equations.

19 The theory: Stochastic Structural Stability Theory (S3T) Consider the first two equal-time cumulants: U(x, y, t) = Z(x, y, t) = D E u(x, y, t) D E (x, y, t) 0 (x, y, t) = (x, y, t) Z(x, y, t) D E C(x 1,x 2,y 1,y 2,t)= 0 (x 1,y 1,t) 0 (x 2,y 2,t) h i = ensemble average over realizations of the excitation

20 Ergodic assumption h i = zonal average of a zonally unbounded single realization of the stochastic excitation Then we have: C(x 1 U(y, t) =u(x, y, t) Z(y, t) = (x, y, t) 0 (x, y, t) = (x, y, t) Z(y, t) D E x 2,y 1,y 2,t)= 0 (x 1,y 1,t) 0 (x 2,y 2,t)

21 The theory: Stochastic Structural Stability Theory t U = hv 0 0 i ru Q(x 1 x 2,y 1 y 2 )= D E = f(x 1,y 1,t)f(x 2,y t C =(A 1 + A 2 )C + "Q A j (U) = U(y j )@ xj +( U 00 (y j xj 1 j v 0 0 = hv 0 0 i = R(C) r (j =1, 2) QL system U, 0 S3T system U,C ensemble average dynamics of the QL system

22 We have three dynamical systems NL simulation QL simplified simulation S3T theory

23 S3T t U = R(C) t C =(A 1 + A 2 )C + "Q S3T system admits equilibria (U E,C E ) U E =0 and C E = " 2r Q is an equilibrium for all, dissipation values and energy input rates " > 0 r>0

24 Eddies tend to reinforce zonal flow inhomogenuities broadband forcing broadband forcing β =0, ε = broadband forcing broadband forcing β =0, ε = v 0 0 U 5 4 y 3 y ( ) δu,( ) δ v ζ x ( ) δu,( ) δ v ζ x 10 6

25 Stability of S3T equilibria perturbing the S3T equilibrium (U E + U, C E + t U = R( C) r t C =(A 1 + A 2 ) C +( A 1 + A 2 )C E with A j = A j (U E + U) A j (U E ), j =1, 2

26 Stability of S3T equilibria perturbing the S3T equilibrium (U E + U, C E + t U = R( C) r t C =(A 1 + A 2 ) C +( A 1 + A 2 )C E with A j = A j (U E + U) A j (U E ), j =1, 2 searching for eigensolutions ( U, C) =( Û, Ĉ)e t Û Ĉ = L Û Ĉ L = L(U E,C E ) (for N y = 128 ) dim(l) )

27 Stability of S3T homogeneous equilibrium for the homogeneous equilibrium U E =0 eigensolutions are harmonics: Û = e iny NIF n x isotropic IRFnforcing n C E = " 2r Q (contours of Re( )),

28 Comparison of S3T predictions with NL dynamics 6 (a) NL U (y, t), ε/ε c = (c) NL vs S3T 10 3 y S3T QL NL E m NL t (b) S3T U (y, t), ε/ε c = y S3T t y isotropic forcing = 71, t " = = 10 " c U

29 S3T predicts jet formation bifurcation 0.7 zmf = E m/(e m + E p) NL S3T 0 ε E m : zonal energy, E p : eddy energy

30 Stability of S3T equilibria Stability analysis of the ideal states predicts: NIF at k =1,...,14, r =10 1, rm =10 2, β = subcritical bifurcation formation of jets existence of multiple equilibria and their domain of attraction merging of jets ϵ/ϵc supercritical bifurcation n

31 Stability of S3T equilibria For higher energy input rates equilibria become S3T unstable and move towards the left of the diagram NIF at k =1,...,14, r =10 1, rm =10 2, β =10 ϵ/ϵc subcritical bifurcation S3T equilibria (stable & unstable) are hydrodynamically stable supercritical bifurcation n

32 Conclusions QL dynamics captures the jet formation process - The turbulent state is essentially determined by a wave/mean flow interaction S3T provides a closure of this turbulent system and a theory for the emergence, equilibration and the structural stability of the associated turbulent equilibria S3T introduces a new concept of instability arising from the interaction between turbulence with the large scale flow S3T predicts: the formation of jets as an eddy/mean flow S3T instability the existence of multiple equilibria as climate states and their stability jet merger dynamics

33 Furthermore... A Ergodic assumption h i = Reynolds average over an intermediate time scale It is possible to obtain non-zonal and even traveling wave finite amplitude S3T equilibria Bakas and Ioannou, 2013: Emergence of large scale structure in barotropic beta-plane turbulence. Phys. Rev. Lett. 110,

34 Generalized S3T equilibria generalized S3T admits equilibria with zonal as well as non-zonal spectral components

35 B S3T applied to wall-bounded shear flow y Formation of roll/streak structures in wall-bounded Couette/Poisseuille flow can be identified as ST3 instability U Couette flow + homogeneous turbulence a test function perturbation streak induces a supporting roll Farrell and Ioannou, 2012: Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, Constantinou et. al., 2013: Turbulence in the restricted dynamics of the S3T/RNL system: comparison with DNS. J. Phys. Conf. Ser. (to appear). y z

36 Ευχαριστώ This work has been supported by Constantinou, N.C., Farrell, B. F. and Ioannou, P.J., 2013: Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory. J. Atmos. Sci., doi: /jas-d , in press.